which distinguishes the two types of dynamical transitions. We observe that for type-I transition, the winding number \(N_{\mathrm {E}}=0\)  is trivial, while the winding \(N_{\mathrm {E}}=1\)  is nontrivial for the type-II dynamical transition. Consequently, these distinct exceptional points on dBISs shows the fundamental difference between the type-I and type-II dynamical transitions. Moreover, regardless of the shape and size of the loop \(\mathcal {S}\) , the winding number \(N_{\mathrm {E}}\)  only depends on the topological properties of the enclosed exceptional points as long as the loop does not cross any other singular points; see Figs. REF a and REF b. Here we note that the topological charges are always singularities of the field \((\langle L_{x}\rangle ,\langle L_{y}\rangle )\)  and have nontrivial winding number [1]} [see Figs. REF c and REF d]. The loop \(\mathcal {S}\)  should be introduced without enclosing any non-exceptional charge momentum in characterizing the dynamical transitions and corresponding exceptional points. This also tells that the type-II dynamical transition is similar to the equilibrium topological phase transition, in which the topological charges serve as singular points and the transition occurs when they pass through the BISs [2]}, [3]}. On the other hand, the type-I transition is a peculiar feature of the quench-induced non-Hermitian dynamical topological phase transition.
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